To find the polynomial $f(x)$ of degree 4 with the given zeros $3, -4, 7, 0$, we use the fact that the polynomial can be expressed in factored form as:

$f(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)$

where $r_1, r_2, r_3, r_4$ are the roots (or zeros) of the polynomial, and $a$ is a constant.

Step 1: Write the factors for the given zeros

Given zeros are $3, -4, 7, 0$, so the factors are: $f(x) = a(x - 3)(x + 4)(x - 7)(x)$

Step 2: Simplify (if needed)

In factored form, the polynomial is already: $f(x) = a(x - 3)(x + 4)(x - 7)(x)$

If no specific value for $a$ is given, you can assume $a = 1$: $f(x) = (x - 3)(x + 4)(x - 7)(x)$

Let me know if you want the expanded form or further explanation!

Do you have any additional questions or need help with the expansion?

Here are 5 related questions you might consider:

1. How do you expand a polynomial like this one?
2. What happens to the graph of $f(x)$ if we change the value of $a$?
3. How can you verify that the given zeros correspond to this polynomial?
4. What is the degree of the polynomial in this form?
5. How do you find the leading coefficient of $f(x)$?

Tip: The product of the zeros determines the constant term (if $a = 1$) in the expanded form.